Does the Integral of Riemman Zeta Function have a meaning?

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Discussion Overview

The discussion revolves around the meaning and properties of the integral of the Riemann zeta function, exploring its theoretical implications and numerical methods for evaluation. The scope includes mathematical reasoning and conceptual clarification regarding the function's behavior in complex analysis.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant suggests using numerical methods to evaluate the integral of the Riemann zeta function and inquires about its meaning.
  • Another participant asks for clarification on which specific integral is being referenced.
  • A third participant proposes a specific integral representation of the Riemann zeta function, highlighting its mathematical beauty.
  • One participant reiterates their interest in numerical methods and suggests that a polynomial could provide values for the integral over any interval, while also stating that the Riemann zeta function is meromorphic with a single pole at 1, allowing for line integrals along curves that do not pass through the pole.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and interest in the integral of the Riemann zeta function, but no consensus is reached regarding its meaning or the specific integral being discussed.

Contextual Notes

There are unresolved assumptions regarding the choice of integral and the implications of using numerical methods. The discussion does not clarify the specific conditions under which the integral is evaluated.

JorgeM
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I have been trying to use numerical methods with this function but now I realize that I if I could suggest a Polynomial in theory, I could get some value for the Integral at least in any interval. In general, does the Integral of the Riemman dseta function has a meaning by itself?
 
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Which integral?
 
Perhaps this one:

$$ \zeta(s)=\frac{\Gamma(1-s)}{2 \pi i} \mathop\int_{\multimap} \frac{x^{s-1}}{e^{-x}-1}dx$$

Nevertheless, one of the most beautiful constructs in Mathematics IMO.
 
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JorgeM said:
I have been trying to use numerical methods with this function but now I realize that I if I could suggest a Polynomial in theory, I could get some value for the Integral at least in any interval. In general, does the Integral of the Riemman dseta function has a meaning by itself?

I think that the Riemann zeta function is meromorphic in the entire complex plane with a single pole at 1 . It makes sense to takes its line integral along any piecewise smooth curve that goes not pass through the pole.
 

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